So just a note, you've got a great deal of repetition in your search. 0.5, for instance, turns up once for every even denominator. That's going to skew your percentages a bit.
Yes, the constructed list contains a lot of duplicates (removing duplicates with the python set operations shortens it by 40% - would be a worthwhile optimization for the runtime, too), but my percentages are not based on that.
But: I sampled a random real, and looked at whether any fraction in the list is close enough to this real. And repeated the sampling 1000 times. This is how the percentages are calculated, and the duplicates in the list don't affect this.
